# On colorful edge triples in edge-colored complete graphs

## TL;DR

This paper studies special edge-colorings of complete graphs that prevent monochromatic subgraphs while ensuring every small subset contains a multicolored copy, providing exact color bounds and exploring related Ramsey capacities.

## Contribution

It determines the minimum number of colors for $F$-caring colorings in complete graphs for key cases and constructs explicit colorings with logarithmic color bounds.

## Key findings

- Exact color bounds for $F$-caring colorings when $F=K_{1,3}$ and $F=P_4$
- Explicit construction of $2 ext{log}_2 n$ colorings with multicolored $P_4$
- Shannon capacity of Gr"otzsch graph exceeds that of a 5-cycle

## Abstract

An edge-coloring of the complete graph $K_n$ we call $F$-caring if it leaves no $F$-subgraph of $K_n$ monochromatic and at the same time every subset of $|V(F)|$ vertices contains in it at least one completely multicolored version of $F$. For the first two meaningful cases, when $F=K_{1,3}$ and $F=P_4$ we determine for infinitely many $n$ the minimum number of colors needed for an $F$-caring edge-coloring of $K_n$. An explicit family of $2\lceil\log_2 n\rceil$ $3$-edge-colorings of $K_n$ so that every quadruple of its vertices contains a totally multicolored $P_4$ in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Gr\"otzsch graph is strictly larger than that of the five length cycle.